4 Answers
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The tier system was introduced in dnd-4e, and is a more formal development of ideas from earlier editions.

Heroic tier: Levels 1-10.

Characters may have impressive skills, but operate on a basically human level.

Adventures take place in local environments - dungeons, towns, forests.

Threats are mostly part of the local ecology, or summoned or created. (Natural creatures, other sapient species, created mechanisms, plants.)

Paragon tier: Levels 11-20

Characters now have extreme, near-superhuman levels of their lead skills. They can accomplish things no ordinary human could (and make very difficult skill DC rolls!)

Adventures take place in a wider arena. They may save entire kingdoms, not just local villages. Their growing reputations will make them major players, even if birth and rank don't. They might lead guilds, be involved in court politics, or command soldiers.

Enemies also exist on a larger scale. Extraplanar threats become more common, and less likely to have to be summoned first. Players may meet dragons, invading warlords (and their armies), elemental or demonic creatures, colossal magical beasts.

Characters gain powers from a 'paragon class' - a development of the 'prestige class' idea from D&D 3e. The paragon class gives tightly-focused powers related to a specific concept of how to play the character's main class. (For example: A druid who specialises in driving animals berserk. A warlock who steals life from opponents. A barbarian who becomes more and more like a bear.)

Epic tier: Levels 21-30

Characters can accomplish awesome and impossible things with skills alone, before they even bother to use their class powers. Which are increasingly powerful.

Adventures are routinely extra-planar - if the characters even make their homes on their original world any more - and threats are ancient dragons, powerful planar entities, titans, or the like. Entire worlds or areas of existence may be at stake.

Each character progresses towards an 'epic destiny' - chosen by the player at L21. They gradually gain extra powers appropriate to this destined ending. (For example: becoming a god, or a transcendent energy-entity, or a heroic legend, or an immortal traveller.)

This effectively gives the GM 10 levels notice to plan the character's heroic final fate at level 30, which is where D&D 4e ends.

(The system has developed from a concept present even in very early versions of D&D, that a high-level character would eventually become immortal.
The BECMI D&D is the first version with this idea, providing for immortality after level 36. Later editions had the concept of 'Epic levels', beginning at level 21. This progression tended to be slower than at levels 1-20, but to allow otherwise impossible feats, and continue to immortality. In D&D 3rd edition, Epic levels were 21-40, and Deities and Demigods provided limited rules support for becoming gods at levels 41-60.)

No one playing 3e uses the "tier" nomenclature, it is new and unique to 4e. You could certainly port the concept but that's not really done. 3e had the idea of "epic levels" after level 20, but not heroic/paragon or "tiers".
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mxyzplk♦Apr 18 '11 at 17:18

Generally, the heroic tier is dealing with threats native to the campaign world, Paragon tier is dealing with extraplanar threats, and epic are some other plane's extraplanar threats... but that's a gross simplification.

In Basic levels, characters dungeon crawl. While written modules are levels 1-3, the basic sets all include levels 1-5, and basic modules are often not too easy for level 4 and 5 parties.
In Expert levels, characters travel overland to reach dungeons. They may begin to become political.
in Companion levels, characters establish dominions and become rulers, and begin to travel the multiverse.
In Master levels, characters routinely travel between planes, and begin to quest for immortality.
In Immortal levels, characters are essentially demi-gods... and can rise to being major movers and shakers of the multiverse. The Later Wrath of the Immortals rules change the mechanics, but not the general tiers. There are 6 subtiers in either version, each 6 levels wide, but those subtiers were not made use of on the adventures, unlike the B/X/CM/M tiers (which had separate module lines). Many would argue the term "demigod" for levels i24-i36; they are capable of quite interesting feats of universe alteration.